1-3hit |
Atsushi TAKAHASHI Shuichi UENO Yoji KAJITANI
The family Pk of graphs with proper-path-width at most k is minor-closed. It is known that the number of minimal forbidden minors for a minor-closed family of graphs is finite, but we have few such families for which all the minimal forbidden minors are listed. Although the minimal acyclic forbidden minors are characterized for Pk, all the minimal forbidden minors are known only for P1. This paper lists 36 minimal forbidden minors for P2, and shows that there exist no other minimal forbidden minors for P2.
Atsushi TAKAHASHI Shuichi UENO Yoji KAJITANI
A graph G is said to be universal for a family F of graphs if G contains every graph in F as a subgraph. A minimum universal graph for F is a universal graph for F with the minimum number of edges. This paper considers a minimum universal graph for the family Fkn of graphs on n vertices with path-width at most k. We first show that the number of edges in a universal graph Fkn is at least Ω(kn log(n/k)). Next, we construct a universal graph for Fkn with O(kn log(n/k)) edges, and show that the number of edges in a minimum universal graph for Fkn is Θ(kn log(n/k)) .
Atsushi TAKAHASHI Shuichi UENO Yoji KAJITANI
We introduce the interval set of a graph G which is a representation of the proper-path-decomposition of G, and show a linear time algorithm to construct an optimal interval set for any tree T. It is shown that a proper-path-decomposition of T with optimal width can be obtained from an optimal interval set of T in O(n log n) time.